In this dissertation we employ stack and queue layouts of graphs to explore the relative power of stacks and queues. Stack layout and queue layouts of graphs can be examined from several points of view. A stack or a queue layout of a graph can be thought of as an embedding of the graph in a plane satisfying certain constraints, or as a graph linearization that optimizes certain cost measures, or as a scheme to process the edges of the graph using the fewest number of stacks or queues. All three points of view permeate this research, though the third point of view dominates. Specific problems in stack and queue layouts of graphs have their origin in the areas of VLSI, fault-tolerant computing, scheduling parallel processes, sorting with a network of stacks and queues, and matrix computations. We first present two tools that are useful in the combinatorial and algorithmic analysis of stack and queue layouts as well as in determining bounds on the stacknumber and the queuenumber for a variety of graphs. The first tool is a formulation of a queue layout of a graph as a covering of its adjacent matrix with staircases. Not only does this formulation serve as a tool for analyzing stack and queue layouts, it also leads to efficient algorithms for several problems related to sequences, graph theory, and computational geometry. The connection between queue layouts and matrix covers also forms the basis of a new scheme for performing matrix computations on a data driven network. Our analysis reveals that this scheme uses less hardware and is faster than existing schemes. The second tool is obtained by considering separated and mingled layouts of graphs. This tool allows us to obtain lower bounds on the stacknumber and the queuenumber of a graph by partitioning the graph into subgraphs and simply concentrating on the interaction of the subgraphs. These tools are used to obtain results in three areas. The first area is stack and queue layouts of directed acyclic graphs (dags). This area is motivated by problems of scheduling parallel processes. We establish the stacknumber and the queuenumber of classes of dags such as trees, unicylic graphs, outerplanar graphs, and planar graphs. We then present linear time algorithms to recognize 1-stack dags and leveled-planar dags. In contrast, we show that the problem of recognizing 9-stack dags and the problem of recognizing 4-queue dags are both NP-complete. The second area is stack and queue layouts of partially ordered sets (posets). We establish upper bounds on the queuenumber of a poset in terms of other measures such as length, width, and jumpnumber. We also present lower bounds on the stacknumber and on the queuenumber of certain classes of posets. We conclude by showing that the problem of recognizing a 4-queue poset is NP-complete. The third area is queue layouts of planar graphs. While it has been shown that the stacknumber of the family of planar graphs is 4, the queuenumber of planar graphs is unknown. We conjecture that a family of planar graphs—the stellated triangles—has unbounded queuenumber; using separated and mingled layouts, we demonstrate significant progress towards that result. / Ph. D.